STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE 1125
The distribution function for the normal distribution is
given byFx et tx
() .1
2(^22)
p
∫ d
(12)
It is shown in Figure 4.
The normal distribution is of great importance for any
field which uses statistics. For one thing, it applies where the
distribution is assumed to be the result of a very large number
of independent variable, summed together. This is a common
assumption for errors of measurement, and it is often made
for any variables affected by a large number of random fac-
tors, a common situation in the environmental field.
There are also practical considerations involved in the
use of normal statistics. Normal statistics have been the
most extensively developed for continuous random vari-
ables; analyses involving nonnormal assumptions are apt
to be cumbersome. This fact is also a motivating factor in
the search for transformations to reduce variables which are
described by nonnormal distributions to forms to which the
normal distribution can be applied. Caution is advisable,
however. The normal distribution should not be assumed as
a matter of convenience, or by default, in case of ignorance.
The use of statistics assuming normality in the case of vari-
ables which are not normally distributed can result in serious
errors of interpretation. In particular, it will often result in
the finding of apparent significant differences in hypothesis
testing when in fact no true differences exists.
The equation which describes the density function of the
normal distribution is often found to arise in environmental
work in situations other than those explicitly concerned with
the use of statistical tests. This is especially likely to occur in
connection with the description of the relationship between
variables when the value of one or more of the variables may
be affected by a variety of other factors which cannot be
explicitly incorporated into the functional relationship. For
example, the concentration of emissions from a smokestack
under conditions where the vertical distribution has become
uniform is given by Panofsky as
C
Q
VD
e
y
yy
2
(^222)
ps
s ,
(13)
where y is the distance from the stack, Q is the emission
rate from the stack, D is the height of the inversion layer,
and V is the average wind velocity. The classical diffusion
equation was found to be unsatisfactory to describe this
process because of the large number of factors which can
affect it.
The lognormal distribution is an important non-normal
continuous distribution. It can be arrived at by considering
a theory of elementary errors combined by a multiplicative
process, just as the normal distribution arises out of a theory
of errors combined additively. The probability density func-
tion for the lognormal is given by
fx x
fx
x
exnx
()
() ().
00
1
2
(^120)
22
for
for
ps
ms
(14)
The shape of the lognormal distribution depends on the
values of μ and s 2. Its density function is shown graphically
in Figure 5 for μ = 0, s = 0.5. The positive skew shown is
characteristic of the lognormal distribution.
The lognormal distribution is likely to arise in situa-
tions in which there is a lower limit on the value which
the random variable can assume, but no upper limit. Time
measurements, which may extend from zero to infinity, are
often described by the lognormal distribution. It has been
applied to the distribution of income sizes, to the relative
abundance of different species of animals, and has been
assumed as the underlying distribution for various discrete
counts in biology. As its name implies, it can be normal-
ized by transforming the variable by the use of logarithms.
See Aitchison and Brown (1957) for a further discussion of
the lognormal distribution.
Many other continuous distributions have been studied.
Some of these, such as the uniform distribution, are of minor
–3 –2 –1 (^0123)
0.0
0.2
0.4
0.6
0.8
1.0
F(X)
X(σ UNITS)
FIGURE 4
0
(^246)
0.2
0.4
0.6
f(X)
FIGURE 5
C019_004_r03.indd 1125C019_004_r03.indd 1125 11/18/2005 1:30:56 PM11/18/2005 1:30:56 PM